Existence of a Renormalized Solution of a Quasilinear Elliptic Equation without the Sign Condition on the Lower-Order Term

The paper considers a second-order quasilinear elliptic equation with an integrableright-hand side. Restrictions on the structure of the equation are stated in terms of thegeneralized \(N\) -function. Unlike the author’s previous papers,there is no sign condition on the lower-order term of the equation. The existence of a renormalizedsolution of the Dirichlet problem for this equation is proved in nonreflexiveMusielak–Orlicz–Sobolev spaces in an arbitrary unbounded strictly Lipschitz domain.